U-quadratic distribution: Information from Answers.com

U-Quadratic
Probability density function
Cumulative distribution function
Parameters	$a:~a \in (-\infty,\infty)$ $b:~b \in (a, \infty)$ or $\alpha:~\alpha\in (0,\infty)$ $\beta:~\beta \in (-\infty,\infty),$
Support	$x\in [a , b]\!$
Probability density function (pdf)	$\alpha \left ( x - \beta \right )^2$
Cumulative distribution function (cdf)	${\alpha \over 3} \left ( (x - \beta)^3 + (\beta - a)^3 \right )$
Mean	${a+b \over 2}$
Median	${a+b \over 2}$
Mode	$a and b$
Variance	${3 \over 20} (b-a)^2$
Skewness	$0$
Excess kurtosis	${3 \over 112} (b-a)^4$
Entropy	TBD
Moment-generating function (mgf)	See text
Characteristic function	See text

In probability theory and statistics, the U-quadratic distribution is a continuous probability distribution defined by a unique quadratic function with lower limit a and upper limit b.

$f(x|a,b,\alpha, \beta)=\alpha \left ( x - \beta \right )^2, \quad\text{for } x \in [a , b].$

1 Parameter relations
2 Related distributions
3 Applications
4 Moment generating function
5 Characteristic function
6 Interactive demonstrations
7 External links

Parameter relations

This distribution has effectively only two parameters a, b, as the other two are explicit functions of the support defined by the former two parameters:

$\beta = {b+a \over 2}$

(gravitational balance center, offset), and

$\alpha = {12 \over \left ( b-a \right )^3}$

(vertical scale).

Related distributions

One can introduce a vertically inverted ( $\cap$ )-quadratic distribution in analogous fashion.

Applications

This distribution is a useful model for symmetric bimodal processes. Other continuous distributions allow more flexibility, in terms of relaxing the symmetry and the quadratic shape of the density function, which are enforced in the U-quadratic distribution - e.g., Beta distribution, Gamma distribution, etc.

Moment generating function

$M_X(t) = {-3\left(e^{at}(4+(a^2+2a(-2+b)+b^2)t)- e^{bt} (4 + (-4b + (a+b)^2)t)\right) \over (a-b)^3 t^2 }$

Characteristic function

$\phi_X(t) = {3i\left(e^{iate^{ibt}} (4i - (-4b + (a+b)^2)t)\right) \over (a-b)^3 t^2 }$

Interactive demonstrations

The SOCR tools allow interactive manipulations and computations of the U-quadratic distributions, among other continuous and discrete distributions. Go to SOCR Distributions and select the U-quadraticDistribution from the drop-down list of distributions in this Java applet.

External links

SOCR U-Quadratic Distribution Page

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss-Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule-Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Beta · Irwin–Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Beta prime · Bose–Einstein · Burr · chi-square · chi · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scaled inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (-∞,∞)

Cauchy · extreme value · exponential power · Fisher's z · generalized normal · generalized hyperbolic · Gumbel · hyperbolic secant · Landau · Laplace · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · slash · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · stable · Tweedie

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

	List of mathematics articles (U)
	List of probability distributions
	List of statistics topics

	Why a quadratic equation is called quadratic? Read answer...
	Why quadratic equation called quadratic? Read answer...
	What is a quadrat equation? Read answer...

	Can u Describe the distribution of the erupted materials after the valcano erupts?
	Can u show a picture of a distributer cap for a ford fiesta?
	Sampling Distribution mann-whitney U test?

U-quadratic distribution

Contents